A Characterization of Scoring Rules for Linear Properties

Abstract

We consider the design of proper scoring rules, equivalently proper losses, when the goal is to elicit some function, known as a property, of the underlying distribution. We provide a full characterization of the class of proper scoring rules when the property is linear as a function of the input distribution. A key conclusion is that any such scoring rule can be written in the form of a Bregman divergence for some convex function. We also apply our results to the design of prediction market mechanisms, showing a strong equivalence between scoring rules for linear properties and automated prediction market makers.

Related Material

@InProceedings{pmlr-v23-abernethy12,
title = {A Characterization of Scoring Rules for Linear Properties},
author = {Jacob D. Abernethy and Rafael M. Frongillo},
booktitle = {Proceedings of the 25th Annual Conference on Learning Theory},
pages = {27.1--27.13},
year = {2012},
editor = {Shie Mannor and Nathan Srebro and Robert C. Williamson},
volume = {23},
series = {Proceedings of Machine Learning Research},
address = {Edinburgh, Scotland},
month = {25--27 Jun},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v23/abernethy12/abernethy12.pdf},
url = {http://proceedings.mlr.press/v23/abernethy12.html},
abstract = {We consider the design of proper scoring rules, equivalently proper losses, when the goal is to elicit some function, known as a property, of the underlying distribution. We provide a full characterization of the class of proper scoring rules when the property is linear as a function of the input distribution. A key conclusion is that any such scoring rule can be written in the form of a Bregman divergence for some convex function. We also apply our results to the design of prediction market mechanisms, showing a strong equivalence between scoring rules for linear properties and automated prediction market makers.}
}

%0 Conference Paper
%T A Characterization of Scoring Rules for Linear Properties
%A Jacob D. Abernethy
%A Rafael M. Frongillo
%B Proceedings of the 25th Annual Conference on Learning Theory
%C Proceedings of Machine Learning Research
%D 2012
%E Shie Mannor
%E Nathan Srebro
%E Robert C. Williamson
%F pmlr-v23-abernethy12
%I PMLR
%J Proceedings of Machine Learning Research
%P 27.1--27.13
%U http://proceedings.mlr.press
%V 23
%W PMLR
%X We consider the design of proper scoring rules, equivalently proper losses, when the goal is to elicit some function, known as a property, of the underlying distribution. We provide a full characterization of the class of proper scoring rules when the property is linear as a function of the input distribution. A key conclusion is that any such scoring rule can be written in the form of a Bregman divergence for some convex function. We also apply our results to the design of prediction market mechanisms, showing a strong equivalence between scoring rules for linear properties and automated prediction market makers.

TY - CPAPER
TI - A Characterization of Scoring Rules for Linear Properties
AU - Jacob D. Abernethy
AU - Rafael M. Frongillo
BT - Proceedings of the 25th Annual Conference on Learning Theory
PY - 2012/06/16
DA - 2012/06/16
ED - Shie Mannor
ED - Nathan Srebro
ED - Robert C. Williamson
ID - pmlr-v23-abernethy12
PB - PMLR
SP - 27.1
DP - PMLR
EP - 27.13
L1 - http://proceedings.mlr.press/v23/abernethy12/abernethy12.pdf
UR - http://proceedings.mlr.press/v23/abernethy12.html
AB - We consider the design of proper scoring rules, equivalently proper losses, when the goal is to elicit some function, known as a property, of the underlying distribution. We provide a full characterization of the class of proper scoring rules when the property is linear as a function of the input distribution. A key conclusion is that any such scoring rule can be written in the form of a Bregman divergence for some convex function. We also apply our results to the design of prediction market mechanisms, showing a strong equivalence between scoring rules for linear properties and automated prediction market makers.
ER -